Friday, 21 February 2014

Simple Pendulum

 


A particle of mass   m   is attached by a light inextensible string of length     to a fixed point   O.   It oscillates in a vertical plane under the force of gravity through a small angle .

We have to fined the period of oscillation .

Let    be the point of suspension, 
OA   the vertical .
At any time  t,   let   P  be the position of the particle,
Angle AOP   being   theta redians    and
arc AP   being   s.

The force that act on the particle are its weight   mg  vertically downwards and the tension of the string along   PO.

Resolving along the   tangent PQ  to the circle

m(d/dt)(ds/dt)=-mg sin theta   ------ (1)

The amplitude of the oscillation is small and

therefore  sin theta=theta

since ;  s=theta l ,

equation  (1)  reduces to

(d/dt)(ds/dt)=gs/l      -------------- (2)

Thus the motion is simple harmonic and the period of a Complete Oscillation is

2 pai multiplied by under root of (l/g)



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